We will learn how to express of a simple quadratic surd. We cannot express a simple quadratic surd by the following ways:
I. A simple quadratic surd cannot be equal to the sum or difference of a rational quantity and a simple quadratic surd.
Suppose, let √p a given quadratic surd.
If possible, let us assume, √p = m + √n where m is a rational quantity and √n is a simple quadratic surd.
Now, √p = m + √n
Squaring both sides, we get,
p = m^2 + 2m√n + n
m^2 +2m√n + n = p
2m√n = p  m^2  n
√m = (p  m^2  n)/2m, which is a rational quantity.
From the above expression we can clearly see that the value
of a quadratic surd is equal to a rational quantity which is impossible.
Similarly, we can prove that √p ≠ m  √n
Therefore, the value of a simple quadratic surd cannot be equal to the sum or difference of a rational quantity and a simple quadratic surd.
II. A simple quadratic surd cannot be equal to the sum or difference of two simple unlike quadratic surds.
Suppose, let √p be a given simple quadratic surd. If possible, let us assume √p = √m + √n are two simple quadratic surds.
Now, √p = √m + √n
Squaring both sides we get,
p = m + 2√mn + n
√mn = (p  m  n)/2, which is a rational quantity.
From the above expression we can clearly see that the value of a quadratic surd is equal to a rational quantity, which is obviously impossible, since √m and √n are two unlike quadratic surds, hence √m ∙ √n = √mn cannot be rational.
Similarly, our assumption cannot be correct i.e. √p = √m + √n does not hold.
Similarly, we can prove that, √p ≠ √m  √n.
Therefore, the value of a simple quadratic surd cannot be equal to the sum or difference of two simple unlike quadratic surds.
`11 and 12 Grade Math
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